OFFSET
1,1
COMMENTS
Theorem: 8x^4-1 and 8x^4+1 can both be prime iff x = 3m for some integer m. Proof: If x != 3m then x=3m+1 or x=3m+2. If x = 3m+1, then 8x^4+1 = 8(81*m^4 + 108*m^3 + 54*m^2 + 12*m)+8+1 = 3H for some H. If x = 3m+2, then 8x^4+1 = 8(81*m^4 + 216*m^3 + 216*m^2 + 96*m)+128+1 = 3H for some H. Since 8x^4+1 cannot be prime for x != 3m for all m, it follows that 8x^4-1 and 8x^4+1 can both be prime only if x = 3m for some m. A proof that this sequence is infinite would be good to have.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
For k = 75, 8*k^4 - 1 = 253124999 is prime, 8*k^4 + 1 = 253125001 is prime so 253125001 is the first entry.
MATHEMATICA
Select[648 * Range[1000]^4 + 1, And @@ PrimeQ[# - {0, 2}] &] (* Amiram Eldar, Dec 26 2019 *)
PROG
(PARI) twin8k3(n) = {local(a, b, c, x); c=0; forstep(x=3, n, 3, a=8*x^4-1; b=8*x^4+1; if(ispseudoprime(a)&ispseudoprime(b), c++; print1(b", "); ); ); print(); print(c) }
CROSSREFS
KEYWORD
nonn
AUTHOR
Cino Hilliard, Jul 31 2006
EXTENSIONS
Offset corrected and more terms added by Amiram Eldar, Dec 26 2019
STATUS
approved